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Full text of "Men Of Mathematics"

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F MATHEMATICS obviousness of this simple assumed equality, V2 x Vs = y^ taken for granted in school arithmetic, is evident if we visualize what the equality implies: the 'lawless' square roots of 2, 3, Q are to be extracted, the first two of these are then to be multi- plied together, and the result is to come out equal to thte third, As not one of these three roots can be extracted exactly > no matter to how many decimal places the computation is carried, it is clear that the verification by multiplication as just de- scribed will never be complete. The whole human race toiling incessantly through all its existence could never prove in this way that ,V2 X Vs = V6. Closer and closer approximations to equality would be attained as time went on, but finality would continue to recede. To make these concepts of 'approxi- mation' and 'equality' precise, or to replace our first crude conceptions of irrationals by sharper descriptions which -mil obviate the difficulties indicated, was the task Dedekind set himself in the early 1870's - his work on Continuity and Irrational Numbers was published in 1872. The heart of Dedeldnd's theory of irrational numbers is his concept of the 'cut1 or 'section' (Schnitt): a cut separates oB rational numbers into two classes, so that each number in the first class is less than each number in the second class; every such cut which does not 'correspond' to a rational number 'defines* an irrational number. This bald statement needs elaboration, particularly as even an accurate exposition conceals certain subtle difficulties rooted in the theory of the mathematical infinite, which will reappear when we consider the life of Dedekind's friend Cantor. Assume that some rule has been prescribed which separates all rational numbers into ftao classes, say an 'upper' class and a 'lower' class, such that each number in the lower class is less than every number in the upper class. (Such an assumption would not pass unchallenged to-day by all schools of mathematical philosophy. However, for the moment, it may be regarded as unobjectionable.) On this assumption one of three mutually exclusive situations is possible. (A) There may be a number in the lower class which is greatet than every other number in that class. 574